Lecture Notes on Quantum Mechanics: Measurement, Entanglement, and Density Operators

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February 5, 2025

Introduction

This lecture addresses the pivotal concept of measurement in quantum mechanics, contrasting it with classical measurement and emphasizing its unique implications. We begin by defining observables as Hermitian operators, whose eigenvalues represent the possible outcomes of measurement. The probabilistic nature of quantum measurement is explored, leading to the concept of wave function collapse and interpretations such as the Many-Worlds Interpretation.

We then generalize the discussion to projective measurements and Positive Operator Valued Measurements (POVMs), highlighting their role in distinguishing quantum states. Statistical aspects of measurement, including average values and dispersion, are also examined.

Following the discussion of measurement, we address the concept of phase in quantum mechanics, differentiating between global and relative phases and their physical significance. A significant portion of this lecture is dedicated to quantum entanglement, a core concept in quantum computing. We define and explore the properties of entanglement, discussing its implications, including non-locality and its connection to Bell’s inequalities. Superdense coding is presented as a key application of entanglement.

Finally, we introduce the density operator as a tool to describe statistical ensembles and situations with incomplete knowledge of the quantum state, and discuss its time evolution, which is crucial for understanding open quantum systems and statistical mechanics in quantum settings.

Measurement in Quantum Mechanics

Observables and Measurement Outcomes

In classical physics, measurement is considered a passive observation that reveals pre-existing properties of a system without altering it. Quantum mechanics, however, presents a fundamentally different picture. The act of measurement is not merely passive revelation but an active intervention that fundamentally affects the quantum system.

In quantum mechanics, physical quantities, known as observables, are represented by Hermitian operators. These operators, acting on the Hilbert space of the system, have real eigenvalues.

When an observable $O$, associated with a Hermitian operator $\hat{O}$, is measured on a quantum system, the possible outcomes are limited to the eigenvalues $\{\lambda_i\}$ of $\hat{O}$. If $\lambda_i$ is an eigenvalue, then a measurement can yield the result $\lambda_i$.

Probabilistic Nature of Measurement

Quantum measurement is inherently probabilistic. Even if a system is prepared in an identical state multiple times, measurements of the same observable may yield different outcomes. The probability of obtaining a specific eigenvalue is determined by the system’s quantum state prior to measurement.

If a system is in a state $\ensuremath{\left|{\psi}\right\rangle}$ and the observable $\hat{O}$ has an eigenbasis $\{\ensuremath{\left|{e_i}\right\rangle}\}$ such that $\hat{O}\ensuremath{\left|{e_i}\right\rangle} = \lambda_i \ensuremath{\left|{e_i}\right\rangle}$, the probability $P(\lambda_i)$ of measuring the eigenvalue $\lambda_i$ is given by the squared magnitude of the projection of $\ensuremath{\left|{\psi}\right\rangle}$ onto the eigenvector $\ensuremath{\left|{e_i}\right\rangle}$. Expanding $\ensuremath{\left|{\psi}\right\rangle}$ in the eigenbasis of $\hat{O}$ as $\ensuremath{\left|{\psi}\right\rangle} = \sum_i c_i \ensuremath{\left|{e_i}\right\rangle}$, the probability is: \[P(\lambda_i) = |c_i|^2 = |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2\] This equation highlights the probabilistic nature of quantum measurements, a stark contrast to the deterministic nature of classical measurements.

Wave Function Collapse: State Reduction

Upon obtaining a measurement outcome, the quantum system undergoes an abrupt change of state known as wave function collapse or state reduction. This process describes the transition from a superposition of states to a single eigenstate of the measured observable.

Consider an electron with spin initially in a superposition state, e.g., prepared with spin along the x-axis: $\ensuremath{\left|{\psi}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$ in the z-basis. If a measurement of the spin along the z-axis is performed and the outcome corresponds to spin up (eigenvalue associated with $\ensuremath{\left|{0}\right\rangle}$), the state of the electron immediately after the measurement is no longer $\ensuremath{\left|{\psi}\right\rangle}$ but becomes $\ensuremath{\left|{0}\right\rangle}$. The superposition is eliminated, and the wave function is said to have collapsed to the eigenstate corresponding to the observed eigenvalue.

Mathematically, if a measurement of observable $\hat{O}$ yields eigenvalue $\lambda_i$, and the pre-measurement state is $\ensuremath{\left|{\psi}\right\rangle}$, the post-measurement state $\ensuremath{\left|{\psi'}\right\rangle}$ is the projection of $\ensuremath{\left|{\psi}\right\rangle}$ onto the eigenspace of $\hat{O}$ corresponding to $\lambda_i$, normalized to unity: \[\ensuremath{\left|{\psi'}\right\rangle} = \frac{P_i \ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}P_i \ensuremath{\left|{\psi}\right\rangle}}}\] where $P_i$ is the projection operator onto the eigenspace associated with $\lambda_i$. This collapse is instantaneous, regardless of spatial separation, a feature that raises profound questions about locality in quantum mechanics, especially in the context of entanglement.

Interpretations of Measurement: The Many-Worlds Interpretation

The concept of wave function collapse is conceptually challenging and has led to various interpretations of quantum mechanics. One notable interpretation is the Many-Worlds Interpretation (MWI), proposed by Hugh Everett III.

MWI posits that there is no wave function collapse. Instead, every quantum measurement causes the universe to split into multiple universes. Each possible outcome of the measurement is realized in a separate universe.

In the spin measurement example, upon measurement, the universe splits. In one branch, the electron is spin up and the observer registers spin up; in another branch, the electron is spin down and a copy of the observer registers spin down. Thus, all potential outcomes are actualized, each in its own universe, eliminating the need for wave function collapse.

From the perspective of an observer within a single universe, it appears as if collapse has occurred. However, MWI describes a deterministic evolution of the universal wave function, consistent with the unitary evolution of quantum mechanics, without requiring a separate postulate for measurement. This interpretation remains a subject of ongoing debate and discussion.

Projective Measurements

Projective measurements are a specific type of quantum measurement commonly used in quantum computation and quantum information. They are described by a set of projection operators $\{P_m\}$ that are orthogonal and complete, i.e., $\sum_m P_m = I$ and $P_m P_{m'} = \delta_{mm'} P_m$.

For instance, measurement in the computational basis $\{\ensuremath{\left|{0}\right\rangle}, \ensuremath{\left|{1}\right\rangle}\}$ is a projective measurement. The projection operators are $P_0 = \ensuremath{\left|{0}\middle\rangle\middle\langle{0}\right|}$ and $P_1 = \ensuremath{\left|{1}\middle\rangle\middle\langle{1}\right|}$. These satisfy the completeness relation $P_0 + P_1 = I$ and orthogonality $P_0 P_1 = P_1 P_0 = 0$.

When a projective measurement is performed on a state $\ensuremath{\left|{\psi}\right\rangle}$, the probability of obtaining outcome $m$ is $P(m) = \ensuremath{\left\langle{\psi}\right|}P_m\ensuremath{\left|{\psi}\right\rangle}$, and if outcome $m$ is obtained, the state collapses to $\ensuremath{\left|{\psi'_m}\right\rangle} = \frac{P_m\ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}P_m\ensuremath{\left|{\psi}\right\rangle}}}$.

General Quantum Measurements: Measurement Operators and POVMs

A more general description of quantum measurements is given by measurement operators $\{M_m\}$. Each operator $M_m$ corresponds to a possible measurement outcome $m$.

Measurement Operators

For a measurement described by operators $\{M_m\}$, the probability of obtaining outcome $m$ when measuring a state $\ensuremath{\left|{\psi}\right\rangle}$ is: \[P(m) = \ensuremath{\left\langle{\psi}\right|}M_m^\dagger M_m \ensuremath{\left|{\psi}\right\rangle}\] If outcome $m$ is obtained, the post-measurement state is: \[\ensuremath{\left|{\psi'_m}\right\rangle} = \frac{M_m \ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}M_m^\dagger M_m \ensuremath{\left|{\psi}\right\rangle}}}\] For these operators to describe a valid measurement, they must satisfy the completeness relation: \[\sum_m M_m^\dagger M_m = I\] This condition ensures that the probabilities of all possible outcomes sum to unity.

Positive Operator Valued Measurements (POVMs)

Defining $E_m = M_m^\dagger M_m$, we obtain a set of positive operators $\{E_m\}$ that sum to the identity, $\sum_m E_m = I$. This set $\{E_m\}$ is known as a Positive Operator Valued Measurement (POVM).

A POVM is a set of positive semi-definite operators $\{E_m\}$ that sum to the identity, $\sum_m E_m = I$. The probability of obtaining outcome $m$ for a state $\ensuremath{\left|{\psi}\right\rangle}$ is $P(m) = \ensuremath{\left\langle{\psi}\right|}E_m\ensuremath{\left|{\psi}\right\rangle}$.

POVMs provide the most general framework for describing quantum measurements, encompassing projective measurements as a special case where $M_m = P_m$ are projection operators. POVMs are particularly relevant in quantum state discrimination and quantum communication, especially when dealing with non-orthogonal states.

Distinguishing Non-Orthogonal Quantum States

Distinguishing between quantum states through measurement is a fundamental task in quantum information. If states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$ are orthogonal, i.e., $\ensuremath{\left\langle{\psi_i}\middle|{\psi_j}\right\rangle} = \delta_{ij}$, they can be perfectly distinguished using projective measurements. For example, $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$ are orthogonal and can be perfectly distinguished by measuring in the computational basis.

However, if states are non-orthogonal, perfect discrimination is generally impossible in a single measurement. Consider $\ensuremath{\left|{\psi_1}\right\rangle} = \ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$. These are not orthogonal. Measuring in the computational basis and obtaining outcome $\ensuremath{\left|{0}\right\rangle}$ does not definitively distinguish between $\ensuremath{\left|{\psi_1}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle}$.

POVMs can be designed to optimize the probability of correctly distinguishing non-orthogonal states, although perfect discrimination remains unattainable. For instance, to distinguish $\ensuremath{\left|{\psi_1}\right\rangle} = \ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$, we can define POVM elements $E_1 = \ensuremath{\left|{1}\middle\rangle\middle\langle{1}\right|}$, $E_2 = \ensuremath{\left|{-}\middle\rangle\middle\langle{-}\right|}$ (where $\ensuremath{\left|{-}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} - \ensuremath{\left|{1}\right\rangle})$ is orthogonal to $\ensuremath{\left|{\psi_2}\right\rangle} = \ensuremath{\left|{+}\right\rangle}$), and $E_3 = I - E_1 - E_2$. Observing outcome $E_1$ guarantees the state was $\ensuremath{\left|{\psi_2}\right\rangle}$, and outcome $E_2$ guarantees it was $\ensuremath{\left|{\psi_1}\right\rangle}$. Outcome $E_3$ is inconclusive. This example illustrates how POVMs can be tailored for optimal, albeit imperfect, discrimination of non-orthogonal states.

Average and Dispersion of Measurements

Average Value or Expectation Value

In quantum mechanics, repeated measurements of an observable on identically prepared systems generally yield a distribution of outcomes. To characterize the central tendency of these outcomes, we use the concept of the average value or expectation value. This represents the mean value of the observable obtained from a large number of measurements.

For an observable $O$ represented by a Hermitian operator $\hat{O}$, and a system in state $\ensuremath{\left|{\psi}\right\rangle}$, the average value $\langle O \rangle_\psi$ is given by: \[\langle O \rangle_\psi = \sum_i \lambda_i P(\lambda_i)\] where $\{\lambda_i\}$ are the eigenvalues of $\hat{O}$, and $P(\lambda_i) = |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2$ is the probability of obtaining the eigenvalue $\lambda_i$ when measuring the observable in state $\ensuremath{\left|{\psi}\right\rangle}$, with $\{\ensuremath{\left|{e_i}\right\rangle}\}$ being the corresponding eigenvectors of $\hat{O}$.

Equivalently, the expectation value can be computed directly using the operator $\hat{O}$ and the state $\ensuremath{\left|{\psi}\right\rangle}$ as: \[\langle O \rangle_\psi = \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle}\] This equivalence can be shown through the spectral decomposition of $\hat{O}$, $\hat{O} = \sum_i \lambda_i \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|}$: \[\begin{aligned}\langle O \rangle_\psi &= \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle} \\&= \ensuremath{\left\langle{\psi}\right|} \left( \sum_i \lambda_i \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|} \right) \ensuremath{\left|{\psi}\right\rangle} \\&= \sum_i \lambda_i \ensuremath{\left\langle{\psi}\right|} \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|} \ensuremath{\left|{\psi}\right\rangle} \\&= \sum_i \lambda_i \ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle} \ensuremath{\left\langle{\psi}\middle|{e_i}\right\rangle} \\&= \sum_i \lambda_i |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2\end{aligned}\]

Variance and Standard Deviation

To quantify the spread or dispersion of measurement outcomes around the average value, we use the variance and standard deviation. The variance $\text{Var}(O)_\psi$ of an observable $O$ in a state $\ensuremath{\left|{\psi}\right\rangle}$ measures the average squared deviation from the expectation value: \[\text{Var}(O)_\psi = \langle (O - \langle O \rangle_\psi)^2 \rangle_\psi\] Expanding this expression, we obtain a computationally useful form: \[\begin{aligned}\text{Var}(O)_\psi &= \langle O^2 - 2O\langle O \rangle_\psi + \langle O \rangle_\psi^2 \rangle_\psi \nonumber \\&= \langle O^2 \rangle_\psi - 2\langle O \rangle_\psi \langle O \rangle_\psi + \langle O \rangle_\psi^2 \nonumber \\&= \langle O^2 \rangle_\psi - (\langle O \rangle_\psi)^2\end{aligned}\] where $O^2$ represents the operator $(\hat{O})^2$, and $\langle O^2 \rangle_\psi = \ensuremath{\left\langle{\psi}\right|} \hat{O}^2 \ensuremath{\left|{\psi}\right\rangle}$.

The standard deviation $\Delta O_\psi$ is the square root of the variance and provides a measure of the typical deviation of a measurement outcome from the average value, having the same units as the observable $O$: \[\Delta O_\psi = \sqrt{\text{Var}(O)_\psi} = \sqrt{\langle O^2 \rangle_\psi - (\langle O \rangle_\psi)^2}\]

A smaller standard deviation indicates that the measurement outcomes are tightly clustered around the average value, implying a more predictable measurement. Conversely, a larger standard deviation signifies a broader spread of outcomes and less predictability. Notably, if the system is in an eigenstate of $\hat{O}$, the variance and standard deviation are zero, indicating that every measurement will yield the corresponding eigenvalue with certainty, reflecting a dispersionless state for that observable. These statistical measures are crucial for understanding the probabilistic nature of quantum measurements and quantifying the uncertainty inherent in them.

Phase in Quantum Mechanics

Global Phase: Physical Irrelevance

In quantum mechanics, a global phase transformation of a state vector $\ensuremath{\left|{\psi}\right\rangle} \rightarrow e^{i\phi}\ensuremath{\left|{\psi}\right\rangle}$, where $\phi \in \mathbb{R}$, does not alter the physical state. This implies that quantum states are defined up to a global phase factor.

The physical indistinguishability arises because all observable quantities, such as expectation values and transition probabilities, remain invariant under global phase transformations. For any observable $\hat{O}$, the expectation value in the phase-transformed state is: \[\begin{aligned}\langle O \rangle_{e^{i\phi}\psi} &= \ensuremath{\left\langle{e^{i\phi}\psi}\right|} \hat{O} \ensuremath{\left|{e^{i\phi}\psi}\right\rangle} \\&= e^{-i\phi} \ensuremath{\left\langle{\psi}\right|} \hat{O} e^{i\phi} \ensuremath{\left|{\psi}\right\rangle} \\&= e^{-i\phi} e^{i\phi} \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle} \\&= \langle O \rangle_{\psi}\end{aligned}\] Since expectation values, and similarly probabilities, are unchanged, global phase factors have no observable consequences and are thus physically irrelevant.

Relative Phase: Physical Significance

In contrast to global phases, relative phases between the components of a quantum superposition are physically significant. Consider a superposition of two states $\ensuremath{\left|{\psi}\right\rangle} = c_1 \ensuremath{\left|{\psi_1}\right\rangle} + c_2 \ensuremath{\left|{\psi_2}\right\rangle}$. The relative phase is the phase difference between the complex coefficients $c_1$ and $c_2$. Altering this relative phase leads to distinct quantum states with different physical properties.

Example 1 (examplebox, title=Relative Phase Difference). Consider two states formed by superposition of computational basis states: \[\begin{aligned}\ensuremath{\left|{\psi_+}\right\rangle} &= \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle}) \\\ensuremath{\left|{\psi_-}\right\rangle} &= \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} - \ensuremath{\left|{1}\right\rangle}) = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + e^{i\pi}\ensuremath{\left|{1}\right\rangle})\end{aligned}\] While both are superpositions of $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$, they differ by a relative phase of $\pi$ between the coefficients of $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$. These states are orthogonal, $\ensuremath{\left\langle{\psi_+}\middle|{\psi_-}\right\rangle} = 0$, and represent physically distinct states, corresponding to eigenstates of the Pauli X operator $\sigma_x$ with eigenvalues $+1$ and $-1$, respectively. Measurements performed on $\ensuremath{\left|{\psi_+}\right\rangle}$ and $\ensuremath{\left|{\psi_-}\right\rangle}$ will yield different statistical outcomes, demonstrating the physical relevance of relative phase.

Relative phases are fundamental to quantum interference phenomena and are crucial for quantum technologies, including quantum computing and quantum communication. Operations in quantum computing rely heavily on the precise manipulation of relative phases to achieve quantum algorithms and computations.

Entanglement

Composite Systems and Tensor Products

Entanglement is a uniquely quantum phenomenon that arises in composite quantum systems, which are systems composed of two or more subsystems. The description of composite systems requires the mathematical framework of tensor products to combine the Hilbert spaces of individual subsystems.

If system 1 is described by a Hilbert space $V_1$ and system 2 by $V_2$, the composite system 1+2 is described by the tensor product Hilbert space $V = V_1 \otimes V_2$. A product state in this composite space is formed by taking a state $\ensuremath{\left|{\phi_1}\right\rangle} \in V_1$ and a state $\ensuremath{\left|{\phi_2}\right\rangle} \in V_2$ to form $\ensuremath{\left|{\phi_1}\right\rangle} \otimes \ensuremath{\left|{\phi_2}\right\rangle} \in V$. This product state signifies that system 1 is in state $\ensuremath{\left|{\phi_1}\right\rangle}$ and system 2 is in state $\ensuremath{\left|{\phi_2}\right\rangle}$, without any quantum correlation between them. Operators acting on individual subsystems are also extended to the composite system using the tensor product. For example, if $\hat{O}_1$ acts on $V_1$ and $\hat{O}_2$ on $V_2$, their combined action on $V$ is represented by $\hat{O}_1 \otimes \hat{O}_2$.

For instance, a system of two qubits, each with a 2-dimensional Hilbert space, is described by a 4-dimensional Hilbert space $V = V_{\text{qubit 1}} \otimes V_{\text{qubit 2}}$. The computational basis for this composite system is $\{\ensuremath{\left|{00}\right\rangle}, \ensuremath{\left|{01}\right\rangle}, \ensuremath{\left|{10}\right\rangle}, \ensuremath{\left|{11}\right\rangle}\}$, where $\ensuremath{\left|{ij}\right\rangle} \equiv \ensuremath{\left|{i}\right\rangle} \otimes \ensuremath{\left|{j}\right\rangle}$ for $i, j \in \{0, 1\}$.

Definition of Entanglement

A state $\ensuremath{\left|{\Psi}\right\rangle}$ of a composite quantum system is entangled if it cannot be expressed as a product state of the form $\ensuremath{\left|{\Psi}\right\rangle} \neq \ensuremath{\left|{\phi_1}\right\rangle} \otimes \ensuremath{\left|{\phi_2}\right\rangle} \otimes \cdots \otimes \ensuremath{\left|{\phi_n}\right\rangle}$, where $\ensuremath{\left|{\phi_i}\right\rangle}$ belongs to the Hilbert space of the $i$-th subsystem. Entangled states are characterized by quantum correlations between subsystems that are stronger than classical correlations.

States that can be written as product states are termed separable or unentangled. Entanglement is thus a property of quantum states that are not separable.

Entangled States and Quantum Correlations

Entangled states exhibit unique quantum correlations between subsystems. These correlations are fundamentally different from classical correlations and are at the heart of many quantum information applications. A paradigmatic example of entanglement is the Bell state $\ensuremath{\left|{\Phi^+}\right\rangle}$ (or EPR pair) for two qubits: \[\ensuremath{\left|{\Phi^+}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{00}\right\rangle} + \ensuremath{\left|{11}\right\rangle})\] This state is a superposition of two product states, $\ensuremath{\left|{00}\right\rangle}$ and $\ensuremath{\left|{11}\right\rangle}$, and crucially, it cannot be factored into a product of single-qubit states.

In $\ensuremath{\left|{\Phi^+}\right\rangle}$, the qubits are perfectly correlated. If a measurement on the first qubit yields $\ensuremath{\left|{0}\right\rangle}$, a subsequent measurement on the second qubit is guaranteed to also yield $\ensuremath{\left|{0}\right\rangle}$, and similarly for outcome $\ensuremath{\left|{1}\right\rangle}$. This perfect correlation holds instantaneously, even if the qubits are spatially separated, which is a manifestation of quantum non-locality.

Quantum Non-Locality

The correlations inherent in entangled states give rise to quantum non-locality. This concept implies that measurements performed on one subsystem of an entangled pair can instantaneously influence the state of the other subsystem, irrespective of the distance separating them. This apparent instantaneous influence challenges classical notions of locality, which dictate that physical influences cannot propagate faster than the speed of light.

It is crucial to note that while entanglement implies non-locality, it does not permit faster-than-light communication. The outcomes of measurements on individual subsystems are probabilistic, and it is impossible to control these outcomes to transmit specific classical information faster than light. However, non-locality underscores a profound departure from classical physics and highlights the interconnected nature of entangled quantum systems.

Bell’s Theorem and the Rejection of Local Realism

The counter-intuitive nature of entanglement and non-locality prompted investigations into whether quantum mechanics could be explained by a more fundamental, classical-like theory involving hidden variables. These hypothetical variables would predetermine the outcomes of quantum measurements, and the correlations observed in entangled states would arise from these pre-set values, rather than genuine quantum interconnectedness. This viewpoint is known as local realism.

Bell’s Theorem is a cornerstone in quantum mechanics that demonstrates the incompatibility between quantum mechanics and local realism. Formulated by John Stewart Bell, it provides a theoretical framework to test local realism against quantum mechanics. Bell derived inequalities that set upper limits on the correlations achievable by any local realistic theory. Quantum mechanics predicts violations of these Bell inequalities for entangled states.

Numerous experimental tests, known as Bell tests, have been conducted to compare the predictions of quantum mechanics with the bounds set by Bell inequalities. The results of these experiments consistently violate Bell inequalities and align with quantum mechanical predictions, providing strong evidence against local realism and reinforcing the non-local nature of quantum entanglement. These findings imply that quantum correlations cannot be explained by local hidden variables and that quantum mechanics describes a reality fundamentally different from classical intuition.

Superdense Coding: Exploiting Entanglement for Communication

Entanglement is not only a fundamental feature of quantum mechanics but also a valuable resource for quantum information processing and communication. Superdense coding is a quantum communication protocol that leverages entanglement to transmit two classical bits of information using only one qubit, given that the sender and receiver share a pre-established entangled state.

The superdense coding protocol proceeds as follows:

Entanglement Distribution: Alice and Bob initially share an entangled Bell state, for example, $\ensuremath{\left|{\Phi^+}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{00}\right\rangle} + \ensuremath{\left|{11}\right\rangle})$. Alice possesses the first qubit, and Bob the second.
Encoding Classical Bits: Alice wishes to send two classical bits to Bob. Based on the two bits she intends to transmit (00, 01, 10, or 11), Alice applies a specific quantum operation to her qubit:
- To send "00": Alice applies the Identity operator $I$.
- To send "01": Alice applies the Pauli-X operator $\sigma_x$.
- To send "10": Alice applies the Pauli-Z operator $\sigma_z$.
- To send "11": Alice applies the Pauli-Y operator $\sigma_y$ (or sequentially applies Pauli-X and Pauli-Z, up to a global phase).
Qubit Transmission: Alice sends her qubit to Bob through a quantum channel.
Decoding by Bell Basis Measurement: Bob receives the qubit from Alice. He now possesses both qubits of the entangled pair. Bob performs a measurement in the Bell basis on these two qubits. The outcome of this measurement uniquely identifies which of the four Bell states was created by Alice’s operation. Based on the measured Bell state, Bob can unambiguously decode the two classical bits sent by Alice.

Superdense coding demonstrates the potential of entanglement to enhance communication efficiency, allowing for the transmission of more classical information per qubit than is possible without entanglement. This protocol is a prime example of how quantum resources can outperform classical communication limits.

Density Operator

Introduction to the Density Operator

The density operator, also known as the density matrix, is a central concept in quantum mechanics for describing the statistical state of a quantum system. It is indispensable when dealing with mixed states, which are probabilistic ensembles of pure quantum states, or when there is incomplete knowledge about the system’s quantum state.

In contrast to a pure state, which is described by a single state vector $\ensuremath{\left|{\psi}\right\rangle}$, a mixed state arises from a statistical mixture of pure states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$, each occurring with a probability $p_i$. This situation is common in several scenarios:

Statistical Ensembles: In thermodynamics and statistical mechanics, systems are often described by statistical ensembles due to incomplete knowledge of their microscopic configurations.
Open Quantum Systems: Quantum systems interacting with an environment become entangled with it, and tracing out the environment leads to a mixed state description for the system itself.
Probabilistic Preparation: If a quantum system is prepared in different pure states with certain probabilities, the overall state is mixed.
Subsystems of Entangled Systems: Even if a composite system is in a pure entangled state, the state of a subsystem is generally mixed when the other subsystems are ignored or traced out.

The density operator provides a comprehensive way to represent quantum states, encompassing both pure and mixed states within a unified formalism.

Definition and Properties

For an ensemble of pure states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$ with corresponding probabilities $\{p_i\}$, where $\sum_i p_i = 1$ and $p_i \geq 0$, the density operator $\rho$ is defined as: \[\rho = \sum_{i} p_i \ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}\] For a pure state described by a state vector $\ensuremath{\left|{\psi}\right\rangle}$, the density operator is simply $\rho = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}$.

The density operator possesses several key properties:

Hermiticity: The density operator is Hermitian, $\rho^\dagger =\rho$. This ensures that expectation values of observables are real.
Positivity: The density operator is positive semi-definite, meaning for any state $\ensuremath{\left|{\phi}\right\rangle}$, $\ensuremath{\left\langle{\phi}\right|}\rho\ensuremath{\left|{\phi}\right\rangle} = \sum_{i} p_i |\ensuremath{\left\langle{\phi}\middle|{\psi_i}\right\rangle}|^2 \geq 0$. This reflects the probabilistic nature of quantum states and ensures probabilities are non-negative.
Trace Class and Normalization: The density operator is of trace class with trace equal to one, $\text{Tr}(\rho) = \sum_{i} p_i \text{Tr}(\ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}) = \sum_{i} p_i \ensuremath{\left\langle{\psi_i}\middle|{\psi_i}\right\rangle} = \sum_{i} p_i = 1$. This normalization condition is essential for probabilistic interpretation.
Purity: The purity of a quantum state is quantified by $\text{Tr}(\rho^2)$. For a pure state, $\text{Tr}(\rho^2) = 1$, while for a mixed state, $\text{Tr}(\rho^2) < 1$. Specifically, $\text{Tr}(\rho^2) = \sum_{i,j} p_i p_j \text{Tr}(\ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}\ensuremath{\left|{\psi_j}\middle\rangle\middle\langle{\psi_j}\right|}) = \sum_{i,j} p_i p_j |\ensuremath{\left\langle{\psi_j}\middle|{\psi_i}\right\rangle}|^2$. If $\rho = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}$, then $\rho^2 = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}\ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|} = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|} = \rho$, and $\text{Tr}(\rho^2) = \text{Tr}(\rho) = 1$.

The expectation value of an observable $\hat{O}$ for a system described by the density operator $\rho$ is given by: \[\langle O \rangle_\rho = \text{Tr}(\rho \hat{O}) = \text{Tr}(\hat{O} \rho) = \sum_{i} p_i \ensuremath{\left\langle{\psi_i}\right|} \hat{O} \ensuremath{\left|{\psi_i}\right\rangle}\] This trace formula provides a general method to calculate expectation values for both pure and mixed states.

Time Evolution: Liouville-von Neumann Equation

The time evolution of the density operator, for a closed quantum system, is governed by the Liouville-von Neumann equation (also known as the quantum master equation in this unitary case). If the system’s dynamics are dictated by a Hamiltonian operator $\hat{H}$, the time evolution of the density operator $\rho(t)$ is described by:

The Liouville-von Neumann equation describes the time evolution of the density operator $\rho$ for a closed quantum system with Hamiltonian $\hat{H}$: \[i\hbar \frac{d\rho}{dt} = [\hat{H}, \rho] = \hat{H}\rho - \rho\hat{H}\]

Alternatively, using the unitary time evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$, the density operator at time $t$ can be expressed in terms of its initial state $\rho(0)$ as: \[\rho(t) = \hat{U}(t) \rho(0) \hat{U}^\dagger(t)\] where $\rho(0)$ is the density operator at time $t=0$.

For a pure state initially described by $\rho(0) = \ensuremath{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|}$, its evolution under $\hat{U}(t)$ yields $\rho(t) = \hat{U}(t) \ensuremath{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|} \hat{U}^\dagger(t) = \ensuremath{\left|{\hat{U}(t)\ensuremath{\left|{\psi_0}\right\rangle}}\middle\rangle\middle\langle{\ensuremath{\left\langle{\psi_0}\right|}\hat{U}^\dagger(t)}\right|} = \ensuremath{\left|{\psi(t)}\middle\rangle\middle\langle{\psi(t)}\right|}$, where $\ensuremath{\left|{\psi(t)}\right\rangle} = \hat{U}(t) \ensuremath{\left|{\psi_0}\right\rangle}$. This shows that the Schrödinger equation for pure states is recovered as a special case within the density operator formalism. The Liouville-von Neumann equation and the density operator formalism are essential for describing the quantum dynamics of both pure and mixed states, particularly in scenarios involving statistical ensembles or open quantum systems.

Conclusion

This lecture has provided a comprehensive overview of key concepts in quantum mechanics, starting with the unique nature of quantum measurement and its various interpretations. We explored the probabilistic nature of measurement, the phenomenon of wave function collapse, and interpretations such as the Many-Worlds Interpretation. The discussion extended to different types of measurements, including projective measurements and POVMs, and their role in distinguishing quantum states. We also examined the statistical properties of measurement outcomes, focusing on averagevalues and dispersion.

Further, we elucidated the concept of phase in quantum mechanics, distinguishing between the physical irrelevance of global phase and the significance of relative phases. A substantial portion of the lecture was dedicated to quantum entanglement, defining it as a uniquely quantum correlation with implications for non-locality, as evidenced by Bell’s Theorem. Superdense coding was presented as a practical application showcasing the power of entanglement in quantum communication. Finally, we introduced the density operator as a powerful tool for describing mixed quantum states and statistical ensembles, and discussed its time evolution under the Liouville-von Neumann equation. This collective understanding forms a robust foundation for further exploration into advanced quantum topics and technologies.

Key Takeaways:

Quantum measurement is probabilistic and fundamentally alters the state of the system, leading to wave function collapse.
Entanglement represents non-classical correlations that defy classical explanations and underpin many quantum technologies.
Relative phase is a physically significant quantity in quantum mechanics, crucial for interference and quantum computation.
The density operator provides a general framework for describing both pure and mixed quantum states and their time evolution.

Further Topics: Building upon the concepts introduced in this lecture, future discussions will delve into:

A more in-depth analysis of various interpretations of quantum measurement and their implications.
Detailed exploration of Bell’s Theorem, Bell inequalities, and experimental validations of quantum non-locality.
Applications of entanglement in advanced quantum computing algorithms and quantum cryptography protocols.
Quantum operations and quantum channels within the density operator formalism, essential for understanding quantum information processing.
Open quantum systems and decoherence, utilizing the density operator framework to describe realistic quantum systems interacting with their environment.

These forthcoming topics will expand on the foundational knowledge established here, further illuminating the profound and transformative nature of quantum mechanics and its applications in quantum technologies.

--- title: "Lecture Notes on Quantum Mechanics: Measurement, Entanglement, and Density Operators" author: "Your Name" date: "2025-02-05" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture addresses the pivotal concept of measurement in quantum mechanics, contrasting it with classical measurement and emphasizing its unique implications. We begin by defining observables as Hermitian operators, whose eigenvalues represent the possible outcomes of measurement. The probabilistic nature of quantum measurement is explored, leading to the concept of wave function collapse and interpretations such as the Many-Worlds Interpretation. We then generalize the discussion to projective measurements and Positive Operator Valued Measurements (POVMs), highlighting their role in distinguishing quantum states. Statistical aspects of measurement, including average values and dispersion, are also examined. Following the discussion of measurement, we address the concept of phase in quantum mechanics, differentiating between global and relative phases and their physical significance. A significant portion of this lecture is dedicated to quantum entanglement, a core concept in quantum computing. We define and explore the properties of entanglement, discussing its implications, including non-locality and its connection to Bell's inequalities. Superdense coding is presented as a key application of entanglement. Finally, we introduce the density operator as a tool to describe statistical ensembles and situations with incomplete knowledge of the quantum state, and discuss its time evolution, which is crucial for understanding open quantum systems and statistical mechanics in quantum settings. # Measurement in Quantum Mechanics ## Observables and Measurement Outcomes In classical physics, measurement is considered a passive observation that reveals pre-existing properties of a system without altering it. Quantum mechanics, however, presents a fundamentally different picture. The act of measurement is not merely passive revelation but an active intervention that fundamentally affects the quantum system. In quantum mechanics, physical quantities, known as *observables*, are represented by *Hermitian operators*. These operators, acting on the Hilbert space of the system, have real eigenvalues. ::: tcolorbox When an observable $O$, associated with a Hermitian operator $\hat{O}$, is measured on a quantum system, the possible outcomes are limited to the eigenvalues $\{\lambda_i\}$ of $\hat{O}$. If $\lambda_i$ is an eigenvalue, then a measurement can yield the result $\lambda_i$. ::: ## Probabilistic Nature of Measurement Quantum measurement is inherently probabilistic. Even if a system is prepared in an identical state multiple times, measurements of the same observable may yield different outcomes. The probability of obtaining a specific eigenvalue is determined by the system's quantum state prior to measurement. If a system is in a state $\ensuremath{\left|{\psi}\right\rangle}$ and the observable $\hat{O}$ has an eigenbasis $\{\ensuremath{\left|{e_i}\right\rangle}\}$ such that $\hat{O}\ensuremath{\left|{e_i}\right\rangle} = \lambda_i \ensuremath{\left|{e_i}\right\rangle}$, the probability $P(\lambda_i)$ of measuring the eigenvalue $\lambda_i$ is given by the squared magnitude of the projection of $\ensuremath{\left|{\psi}\right\rangle}$ onto the eigenvector $\ensuremath{\left|{e_i}\right\rangle}$. Expanding $\ensuremath{\left|{\psi}\right\rangle}$ in the eigenbasis of $\hat{O}$ as $\ensuremath{\left|{\psi}\right\rangle} = \sum_i c_i \ensuremath{\left|{e_i}\right\rangle}$, the probability is: $$P(\lambda_i) = |c_i|^2 = |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2$$ This equation highlights the probabilistic nature of quantum measurements, a stark contrast to the deterministic nature of classical measurements. ## Wave Function Collapse: State Reduction Upon obtaining a measurement outcome, the quantum system undergoes an abrupt change of state known as *wave function collapse* or *state reduction*. This process describes the transition from a superposition of states to a single eigenstate of the measured observable. Consider an electron with spin initially in a superposition state, e.g., prepared with spin along the x-axis: $\ensuremath{\left|{\psi}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$ in the z-basis. If a measurement of the spin along the z-axis is performed and the outcome corresponds to spin up (eigenvalue associated with $\ensuremath{\left|{0}\right\rangle}$), the state of the electron immediately after the measurement is no longer $\ensuremath{\left|{\psi}\right\rangle}$ but becomes $\ensuremath{\left|{0}\right\rangle}$. The superposition is eliminated, and the wave function is said to have collapsed to the eigenstate corresponding to the observed eigenvalue. Mathematically, if a measurement of observable $\hat{O}$ yields eigenvalue $\lambda_i$, and the pre-measurement state is $\ensuremath{\left|{\psi}\right\rangle}$, the post-measurement state $\ensuremath{\left|{\psi'}\right\rangle}$ is the projection of $\ensuremath{\left|{\psi}\right\rangle}$ onto the eigenspace of $\hat{O}$ corresponding to $\lambda_i$, normalized to unity: $$\ensuremath{\left|{\psi'}\right\rangle} = \frac{P_i \ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}P_i \ensuremath{\left|{\psi}\right\rangle}}}$$ where $P_i$ is the projection operator onto the eigenspace associated with $\lambda_i$. This collapse is instantaneous, regardless of spatial separation, a feature that raises profound questions about locality in quantum mechanics, especially in the context of entanglement. ## Interpretations of Measurement: The Many-Worlds Interpretation The concept of wave function collapse is conceptually challenging and has led to various interpretations of quantum mechanics. One notable interpretation is the *Many-Worlds Interpretation* (MWI), proposed by Hugh Everett III. ::: tcolorbox MWI posits that there is no wave function collapse. Instead, every quantum measurement causes the universe to split into multiple universes. Each possible outcome of the measurement is realized in a separate universe. ::: In the spin measurement example, upon measurement, the universe splits. In one branch, the electron is spin up and the observer registers spin up; in another branch, the electron is spin down and a copy of the observer registers spin down. Thus, all potential outcomes are actualized, each in its own universe, eliminating the need for wave function collapse. From the perspective of an observer within a single universe, it appears as if collapse has occurred. However, MWI describes a deterministic evolution of the universal wave function, consistent with the unitary evolution of quantum mechanics, without requiring a separate postulate for measurement. This interpretation remains a subject of ongoing debate and discussion. ## Projective Measurements *Projective measurements* are a specific type of quantum measurement commonly used in quantum computation and quantum information. They are described by a set of projection operators $\{P_m\}$ that are orthogonal and complete, i.e., $\sum_m P_m = I$ and $P_m P_{m'} = \delta_{mm'} P_m$. For instance, measurement in the computational basis $\{\ensuremath{\left|{0}\right\rangle}, \ensuremath{\left|{1}\right\rangle}\}$ is a projective measurement. The projection operators are $P_0 = \ensuremath{\left|{0}\middle\rangle\middle\langle{0}\right|}$ and $P_1 = \ensuremath{\left|{1}\middle\rangle\middle\langle{1}\right|}$. These satisfy the completeness relation $P_0 + P_1 = I$ and orthogonality $P_0 P_1 = P_1 P_0 = 0$. When a projective measurement is performed on a state $\ensuremath{\left|{\psi}\right\rangle}$, the probability of obtaining outcome $m$ is $P(m) = \ensuremath{\left\langle{\psi}\right|}P_m\ensuremath{\left|{\psi}\right\rangle}$, and if outcome $m$ is obtained, the state collapses to $\ensuremath{\left|{\psi'_m}\right\rangle} = \frac{P_m\ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}P_m\ensuremath{\left|{\psi}\right\rangle}}}$. ## General Quantum Measurements: Measurement Operators and POVMs A more general description of quantum measurements is given by *measurement operators* $\{M_m\}$. Each operator $M_m$ corresponds to a possible measurement outcome $m$. ### Measurement Operators For a measurement described by operators $\{M_m\}$, the probability of obtaining outcome $m$ when measuring a state $\ensuremath{\left|{\psi}\right\rangle}$ is: $$P(m) = \ensuremath{\left\langle{\psi}\right|}M_m^\dagger M_m \ensuremath{\left|{\psi}\right\rangle}$$ If outcome $m$ is obtained, the post-measurement state is: $$\ensuremath{\left|{\psi'_m}\right\rangle} = \frac{M_m \ensuremath{\left|{\psi}\right\rangle}}{\sqrt{\ensuremath{\left\langle{\psi}\right|}M_m^\dagger M_m \ensuremath{\left|{\psi}\right\rangle}}}$$ For these operators to describe a valid measurement, they must satisfy the *completeness relation*: $$\sum_m M_m^\dagger M_m = I$$ This condition ensures that the probabilities of all possible outcomes sum to unity. ### Positive Operator Valued Measurements (POVMs) Defining $E_m = M_m^\dagger M_m$, we obtain a set of positive operators $\{E_m\}$ that sum to the identity, $\sum_m E_m = I$. This set $\{E_m\}$ is known as a *Positive Operator Valued Measurement* (POVM). ::: tcolorbox A POVM is a set of positive semi-definite operators $\{E_m\}$ that sum to the identity, $\sum_m E_m = I$. The probability of obtaining outcome $m$ for a state $\ensuremath{\left|{\psi}\right\rangle}$ is $P(m) = \ensuremath{\left\langle{\psi}\right|}E_m\ensuremath{\left|{\psi}\right\rangle}$. ::: POVMs provide the most general framework for describing quantum measurements, encompassing projective measurements as a special case where $M_m = P_m$ are projection operators. POVMs are particularly relevant in quantum state discrimination and quantum communication, especially when dealing with non-orthogonal states. ## Distinguishing Non-Orthogonal Quantum States Distinguishing between quantum states through measurement is a fundamental task in quantum information. If states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$ are orthogonal, i.e., $\ensuremath{\left\langle{\psi_i}\middle|{\psi_j}\right\rangle} = \delta_{ij}$, they can be perfectly distinguished using projective measurements. For example, $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$ are orthogonal and can be perfectly distinguished by measuring in the computational basis. However, if states are non-orthogonal, perfect discrimination is generally impossible in a single measurement. Consider $\ensuremath{\left|{\psi_1}\right\rangle} = \ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$. These are not orthogonal. Measuring in the computational basis and obtaining outcome $\ensuremath{\left|{0}\right\rangle}$ does not definitively distinguish between $\ensuremath{\left|{\psi_1}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle}$. POVMs can be designed to optimize the probability of correctly distinguishing non-orthogonal states, although perfect discrimination remains unattainable. For instance, to distinguish $\ensuremath{\left|{\psi_1}\right\rangle} = \ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{\psi_2}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle})$, we can define POVM elements $E_1 = \ensuremath{\left|{1}\middle\rangle\middle\langle{1}\right|}$, $E_2 = \ensuremath{\left|{-}\middle\rangle\middle\langle{-}\right|}$ (where $\ensuremath{\left|{-}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} - \ensuremath{\left|{1}\right\rangle})$ is orthogonal to $\ensuremath{\left|{\psi_2}\right\rangle} = \ensuremath{\left|{+}\right\rangle}$), and $E_3 = I - E_1 - E_2$. Observing outcome $E_1$ guarantees the state was $\ensuremath{\left|{\psi_2}\right\rangle}$, and outcome $E_2$ guarantees it was $\ensuremath{\left|{\psi_1}\right\rangle}$. Outcome $E_3$ is inconclusive. This example illustrates how POVMs can be tailored for optimal, albeit imperfect, discrimination of non-orthogonal states. # Average and Dispersion of Measurements ## Average Value or Expectation Value In quantum mechanics, repeated measurements of an observable on identically prepared systems generally yield a distribution of outcomes. To characterize the central tendency of these outcomes, we use the concept of the *average value* or *expectation value*. This represents the mean value of the observable obtained from a large number of measurements. For an observable $O$ represented by a Hermitian operator $\hat{O}$, and a system in state $\ensuremath{\left|{\psi}\right\rangle}$, the average value $\langle O \rangle_\psi$ is given by: $$\langle O \rangle_\psi = \sum_i \lambda_i P(\lambda_i)$$ where $\{\lambda_i\}$ are the eigenvalues of $\hat{O}$, and $P(\lambda_i) = |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2$ is the probability of obtaining the eigenvalue $\lambda_i$ when measuring the observable in state $\ensuremath{\left|{\psi}\right\rangle}$, with $\{\ensuremath{\left|{e_i}\right\rangle}\}$ being the corresponding eigenvectors of $\hat{O}$. Equivalently, the expectation value can be computed directly using the operator $\hat{O}$ and the state $\ensuremath{\left|{\psi}\right\rangle}$ as: $$\langle O \rangle_\psi = \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle}$$ This equivalence can be shown through the spectral decomposition of $\hat{O}$, $\hat{O} = \sum_i \lambda_i \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|}$: $$\begin{aligned}\langle O \rangle_\psi &= \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle} \\&= \ensuremath{\left\langle{\psi}\right|} \left( \sum_i \lambda_i \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|} \right) \ensuremath{\left|{\psi}\right\rangle} \\&= \sum_i \lambda_i \ensuremath{\left\langle{\psi}\right|} \ensuremath{\left|{e_i}\middle\rangle\middle\langle{e_i}\right|} \ensuremath{\left|{\psi}\right\rangle} \\&= \sum_i \lambda_i \ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle} \ensuremath{\left\langle{\psi}\middle|{e_i}\right\rangle} \\&= \sum_i \lambda_i |\ensuremath{\left\langle{e_i}\middle|{\psi}\right\rangle}|^2\end{aligned}$$ ## Variance and Standard Deviation To quantify the spread or *dispersion* of measurement outcomes around the average value, we use the *variance* and *standard deviation*. The variance $\text{Var}(O)_\psi$ of an observable $O$ in a state $\ensuremath{\left|{\psi}\right\rangle}$ measures the average squared deviation from the expectation value: $$\text{Var}(O)_\psi = \langle (O - \langle O \rangle_\psi)^2 \rangle_\psi$$ Expanding this expression, we obtain a computationally useful form: $$\begin{aligned}\text{Var}(O)_\psi &= \langle O^2 - 2O\langle O \rangle_\psi + \langle O \rangle_\psi^2 \rangle_\psi \nonumber \\&= \langle O^2 \rangle_\psi - 2\langle O \rangle_\psi \langle O \rangle_\psi + \langle O \rangle_\psi^2 \nonumber \\&= \langle O^2 \rangle_\psi - (\langle O \rangle_\psi)^2\end{aligned}$$ where $O^2$ represents the operator $(\hat{O})^2$, and $\langle O^2 \rangle_\psi = \ensuremath{\left\langle{\psi}\right|} \hat{O}^2 \ensuremath{\left|{\psi}\right\rangle}$. The *standard deviation* $\Delta O_\psi$ is the square root of the variance and provides a measure of the typical deviation of a measurement outcome from the average value, having the same units as the observable $O$: $$\Delta O_\psi = \sqrt{\text{Var}(O)_\psi} = \sqrt{\langle O^2 \rangle_\psi - (\langle O \rangle_\psi)^2}$$ A smaller standard deviation indicates that the measurement outcomes are tightly clustered around the average value, implying a more predictable measurement. Conversely, a larger standard deviation signifies a broader spread of outcomes and less predictability. Notably, if the system is in an eigenstate of $\hat{O}$, the variance and standard deviation are zero, indicating that every measurement will yield the corresponding eigenvalue with certainty, reflecting a dispersionless state for that observable. These statistical measures are crucial for understanding the probabilistic nature of quantum measurements and quantifying the uncertainty inherent in them. # Phase in Quantum Mechanics ## Global Phase: Physical Irrelevance In quantum mechanics, a global phase transformation of a state vector $\ensuremath{\left|{\psi}\right\rangle} \rightarrow e^{i\phi}\ensuremath{\left|{\psi}\right\rangle}$, where $\phi \in \mathbb{R}$, does not alter the physical state. This implies that quantum states are defined up to a global phase factor. The physical indistinguishability arises because all observable quantities, such as expectation values and transition probabilities, remain invariant under global phase transformations. For any observable $\hat{O}$, the expectation value in the phase-transformed state is: $$\begin{aligned}\langle O \rangle_{e^{i\phi}\psi} &= \ensuremath{\left\langle{e^{i\phi}\psi}\right|} \hat{O} \ensuremath{\left|{e^{i\phi}\psi}\right\rangle} \\&= e^{-i\phi} \ensuremath{\left\langle{\psi}\right|} \hat{O} e^{i\phi} \ensuremath{\left|{\psi}\right\rangle} \\&= e^{-i\phi} e^{i\phi} \ensuremath{\left\langle{\psi}\right|} \hat{O} \ensuremath{\left|{\psi}\right\rangle} \\&= \langle O \rangle_{\psi}\end{aligned}$$ Since expectation values, and similarly probabilities, are unchanged, global phase factors have no observable consequences and are thus physically irrelevant. ## Relative Phase: Physical Significance In contrast to global phases, *relative phases* between the components of a quantum superposition are physically significant. Consider a superposition of two states $\ensuremath{\left|{\psi}\right\rangle} = c_1 \ensuremath{\left|{\psi_1}\right\rangle} + c_2 \ensuremath{\left|{\psi_2}\right\rangle}$. The relative phase is the phase difference between the complex coefficients $c_1$ and $c_2$. Altering this relative phase leads to distinct quantum states with different physical properties. ::: example **Example 1** (examplebox, title=Relative Phase Difference). Consider two states formed by superposition of computational basis states: $$\begin{aligned}\ensuremath{\left|{\psi_+}\right\rangle} &= \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + \ensuremath{\left|{1}\right\rangle}) \\\ensuremath{\left|{\psi_-}\right\rangle} &= \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} - \ensuremath{\left|{1}\right\rangle}) = \frac{1}{\sqrt{2}}(\ensuremath{\left|{0}\right\rangle} + e^{i\pi}\ensuremath{\left|{1}\right\rangle})\end{aligned}$$ While both are superpositions of $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$, they differ by a relative phase of $\pi$ between the coefficients of $\ensuremath{\left|{0}\right\rangle}$ and $\ensuremath{\left|{1}\right\rangle}$. These states are orthogonal, $\ensuremath{\left\langle{\psi_+}\middle|{\psi_-}\right\rangle} = 0$, and represent physically distinct states, corresponding to eigenstates of the Pauli X operator $\sigma_x$ with eigenvalues $+1$ and $-1$, respectively. Measurements performed on $\ensuremath{\left|{\psi_+}\right\rangle}$ and $\ensuremath{\left|{\psi_-}\right\rangle}$ will yield different statistical outcomes, demonstrating the physical relevance of relative phase. ::: Relative phases are fundamental to quantum interference phenomena and are crucial for quantum technologies, including quantum computing and quantum communication. Operations in quantum computing rely heavily on the precise manipulation of relative phases to achieve quantum algorithms and computations. # Entanglement ## Composite Systems and Tensor Products Entanglement is a uniquely quantum phenomenon that arises in *composite quantum systems*, which are systems composed of two or more subsystems. The description of composite systems requires the mathematical framework of tensor products to combine the Hilbert spaces of individual subsystems. If system 1 is described by a Hilbert space $V_1$ and system 2 by $V_2$, the composite system 1+2 is described by the tensor product Hilbert space $V = V_1 \otimes V_2$. A product state in this composite space is formed by taking a state $\ensuremath{\left|{\phi_1}\right\rangle} \in V_1$ and a state $\ensuremath{\left|{\phi_2}\right\rangle} \in V_2$ to form $\ensuremath{\left|{\phi_1}\right\rangle} \otimes \ensuremath{\left|{\phi_2}\right\rangle} \in V$. This product state signifies that system 1 is in state $\ensuremath{\left|{\phi_1}\right\rangle}$ and system 2 is in state $\ensuremath{\left|{\phi_2}\right\rangle}$, without any quantum correlation between them. Operators acting on individual subsystems are also extended to the composite system using the tensor product. For example, if $\hat{O}_1$ acts on $V_1$ and $\hat{O}_2$ on $V_2$, their combined action on $V$ is represented by $\hat{O}_1 \otimes \hat{O}_2$. For instance, a system of two qubits, each with a 2-dimensional Hilbert space, is described by a 4-dimensional Hilbert space $V = V_{\text{qubit 1}} \otimes V_{\text{qubit 2}}$. The computational basis for this composite system is $\{\ensuremath{\left|{00}\right\rangle}, \ensuremath{\left|{01}\right\rangle}, \ensuremath{\left|{10}\right\rangle}, \ensuremath{\left|{11}\right\rangle}\}$, where $\ensuremath{\left|{ij}\right\rangle} \equiv \ensuremath{\left|{i}\right\rangle} \otimes \ensuremath{\left|{j}\right\rangle}$ for $i, j \in \{0, 1\}$. ## Definition of Entanglement ::: tcolorbox A state $\ensuremath{\left|{\Psi}\right\rangle}$ of a composite quantum system is *entangled* if it cannot be expressed as a product state of the form $\ensuremath{\left|{\Psi}\right\rangle} \neq \ensuremath{\left|{\phi_1}\right\rangle} \otimes \ensuremath{\left|{\phi_2}\right\rangle} \otimes \cdots \otimes \ensuremath{\left|{\phi_n}\right\rangle}$, where $\ensuremath{\left|{\phi_i}\right\rangle}$ belongs to the Hilbert space of the $i$-th subsystem. Entangled states are characterized by quantum correlations between subsystems that are stronger than classical correlations. ::: States that can be written as product states are termed *separable* or *unentangled*. Entanglement is thus a property of quantum states that are not separable. ## Entangled States and Quantum Correlations Entangled states exhibit unique quantum correlations between subsystems. These correlations are fundamentally different from classical correlations and are at the heart of many quantum information applications. A paradigmatic example of entanglement is the Bell state $\ensuremath{\left|{\Phi^+}\right\rangle}$ (or EPR pair) for two qubits: $$\ensuremath{\left|{\Phi^+}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{00}\right\rangle} + \ensuremath{\left|{11}\right\rangle})$$ This state is a superposition of two product states, $\ensuremath{\left|{00}\right\rangle}$ and $\ensuremath{\left|{11}\right\rangle}$, and crucially, it cannot be factored into a product of single-qubit states. In $\ensuremath{\left|{\Phi^+}\right\rangle}$, the qubits are perfectly correlated. If a measurement on the first qubit yields $\ensuremath{\left|{0}\right\rangle}$, a subsequent measurement on the second qubit is guaranteed to also yield $\ensuremath{\left|{0}\right\rangle}$, and similarly for outcome $\ensuremath{\left|{1}\right\rangle}$. This perfect correlation holds instantaneously, even if the qubits are spatially separated, which is a manifestation of quantum non-locality. ## Quantum Non-Locality The correlations inherent in entangled states give rise to *quantum non-locality*. This concept implies that measurements performed on one subsystem of an entangled pair can instantaneously influence the state of the other subsystem, irrespective of the distance separating them. This apparent instantaneous influence challenges classical notions of locality, which dictate that physical influences cannot propagate faster than the speed of light. It is crucial to note that while entanglement implies non-locality, it does not permit faster-than-light communication. The outcomes of measurements on individual subsystems are probabilistic, and it is impossible to control these outcomes to transmit specific classical information faster than light. However, non-locality underscores a profound departure from classical physics and highlights the interconnected nature of entangled quantum systems. ### Bell's Theorem and the Rejection of Local Realism The counter-intuitive nature of entanglement and non-locality prompted investigations into whether quantum mechanics could be explained by a more fundamental, classical-like theory involving *hidden variables*. These hypothetical variables would predetermine the outcomes of quantum measurements, and the correlations observed in entangled states would arise from these pre-set values, rather than genuine quantum interconnectedness. This viewpoint is known as *local realism*. ::: tcolorbox **Bell's Theorem** is a cornerstone in quantum mechanics that demonstrates the incompatibility between quantum mechanics and local realism. Formulated by John Stewart Bell, it provides a theoretical framework to test local realism against quantum mechanics. Bell derived inequalities that set upper limits on the correlations achievable by any local realistic theory. Quantum mechanics predicts violations of these Bell inequalities for entangled states. ::: Numerous experimental tests, known as *Bell tests*, have been conducted to compare the predictions of quantum mechanics with the bounds set by Bell inequalities. The results of these experiments consistently violate Bell inequalities and align with quantum mechanical predictions, providing strong evidence against local realism and reinforcing the non-local nature of quantum entanglement. These findings imply that quantum correlations cannot be explained by local hidden variables and that quantum mechanics describes a reality fundamentally different from classical intuition. ## Superdense Coding: Exploiting Entanglement for Communication Entanglement is not only a fundamental feature of quantum mechanics but also a valuable resource for quantum information processing and communication. *Superdense coding* is a quantum communication protocol that leverages entanglement to transmit two classical bits of information using only one qubit, given that the sender and receiver share a pre-established entangled state. The superdense coding protocol proceeds as follows: 1. **Entanglement Distribution:** Alice and Bob initially share an entangled Bell state, for example, $\ensuremath{\left|{\Phi^+}\right\rangle} = \frac{1}{\sqrt{2}}(\ensuremath{\left|{00}\right\rangle} + \ensuremath{\left|{11}\right\rangle})$. Alice possesses the first qubit, and Bob the second. 2. **Encoding Classical Bits:** Alice wishes to send two classical bits to Bob. Based on the two bits she intends to transmit (00, 01, 10, or 11), Alice applies a specific quantum operation to her qubit: - To send \"00\": Alice applies the Identity operator $I$. - To send \"01\": Alice applies the Pauli-X operator $\sigma_x$. - To send \"10\": Alice applies the Pauli-Z operator $\sigma_z$. - To send \"11\": Alice applies the Pauli-Y operator $\sigma_y$ (or sequentially applies Pauli-X and Pauli-Z, up to a global phase). 3. **Qubit Transmission:** Alice sends her qubit to Bob through a quantum channel. 4. **Decoding by Bell Basis Measurement:** Bob receives the qubit from Alice. He now possesses both qubits of the entangled pair. Bob performs a measurement in the Bell basis on these two qubits. The outcome of this measurement uniquely identifies which of the four Bell states was created by Alice's operation. Based on the measured Bell state, Bob can unambiguously decode the two classical bits sent by Alice. Superdense coding demonstrates the potential of entanglement to enhance communication efficiency, allowing for the transmission of more classical information per qubit than is possible without entanglement. This protocol is a prime example of how quantum resources can outperform classical communication limits. # Density Operator ## Introduction to the Density Operator The *density operator*, also known as the density matrix, is a central concept in quantum mechanics for describing the statistical state of a quantum system. It is indispensable when dealing with *mixed states*, which are probabilistic ensembles of pure quantum states, or when there is incomplete knowledge about the system's quantum state. In contrast to a *pure state*, which is described by a single state vector $\ensuremath{\left|{\psi}\right\rangle}$, a mixed state arises from a statistical mixture of pure states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$, each occurring with a probability $p_i$. This situation is common in several scenarios: - **Statistical Ensembles:** In thermodynamics and statistical mechanics, systems are often described by statistical ensembles due to incomplete knowledge of their microscopic configurations. - **Open Quantum Systems:** Quantum systems interacting with an environment become entangled with it, and tracing out the environment leads to a mixed state description for the system itself. - **Probabilistic Preparation:** If a quantum system is prepared in different pure states with certain probabilities, the overall state is mixed. - **Subsystems of Entangled Systems:** Even if a composite system is in a pure entangled state, the state of a subsystem is generally mixed when the other subsystems are ignored or traced out. The density operator provides a comprehensive way to represent quantum states, encompassing both pure and mixed states within a unified formalism. ## Definition and Properties ::: tcolorbox For an ensemble of pure states $\{\ensuremath{\left|{\psi_i}\right\rangle}\}$ with corresponding probabilities $\{p_i\}$, where $\sum_i p_i = 1$ and $p_i \geq 0$, the density operator $\rho$ is defined as: $$\rho = \sum_{i} p_i \ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}$$ For a pure state described by a state vector $\ensuremath{\left|{\psi}\right\rangle}$, the density operator is simply $\rho = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}$. ::: The density operator possesses several key properties: - **Hermiticity:** The density operator is Hermitian, $\rho^\dagger =\rho$. This ensures that expectation values of observables are real. - **Positivity:** The density operator is positive semi-definite, meaning for any state $\ensuremath{\left|{\phi}\right\rangle}$, $\ensuremath{\left\langle{\phi}\right|}\rho\ensuremath{\left|{\phi}\right\rangle} = \sum_{i} p_i |\ensuremath{\left\langle{\phi}\middle|{\psi_i}\right\rangle}|^2 \geq 0$. This reflects the probabilistic nature of quantum states and ensures probabilities are non-negative. - **Trace Class and Normalization:** The density operator is of trace class with trace equal to one, $\text{Tr}(\rho) = \sum_{i} p_i \text{Tr}(\ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}) = \sum_{i} p_i \ensuremath{\left\langle{\psi_i}\middle|{\psi_i}\right\rangle} = \sum_{i} p_i = 1$. This normalization condition is essential for probabilistic interpretation. - **Purity:** The purity of a quantum state is quantified by $\text{Tr}(\rho^2)$. For a pure state, $\text{Tr}(\rho^2) = 1$, while for a mixed state, $\text{Tr}(\rho^2) < 1$. Specifically, $\text{Tr}(\rho^2) = \sum_{i,j} p_i p_j \text{Tr}(\ensuremath{\left|{\psi_i}\middle\rangle\middle\langle{\psi_i}\right|}\ensuremath{\left|{\psi_j}\middle\rangle\middle\langle{\psi_j}\right|}) = \sum_{i,j} p_i p_j |\ensuremath{\left\langle{\psi_j}\middle|{\psi_i}\right\rangle}|^2$. If $\rho = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}$, then $\rho^2 = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|}\ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|} = \ensuremath{\left|{\psi}\middle\rangle\middle\langle{\psi}\right|} = \rho$, and $\text{Tr}(\rho^2) = \text{Tr}(\rho) = 1$. The expectation value of an observable $\hat{O}$ for a system described by the density operator $\rho$ is given by: $$\langle O \rangle_\rho = \text{Tr}(\rho \hat{O}) = \text{Tr}(\hat{O} \rho) = \sum_{i} p_i \ensuremath{\left\langle{\psi_i}\right|} \hat{O} \ensuremath{\left|{\psi_i}\right\rangle}$$ This trace formula provides a general method to calculate expectation values for both pure and mixed states. ## Time Evolution: Liouville-von Neumann Equation The time evolution of the density operator, for a closed quantum system, is governed by the *Liouville-von Neumann equation* (also known as the quantum master equation in this unitary case). If the system's dynamics are dictated by a Hamiltonian operator $\hat{H}$, the time evolution of the density operator $\rho(t)$ is described by: ::: tcolorbox The Liouville-von Neumann equation describes the time evolution of the density operator $\rho$ for a closed quantum system with Hamiltonian $\hat{H}$: $$i\hbar \frac{d\rho}{dt} = [\hat{H}, \rho] = \hat{H}\rho - \rho\hat{H}$$ ::: Alternatively, using the unitary time evolution operator $\hat{U}(t) = e^{-i\hat{H}t/\hbar}$, the density operator at time $t$ can be expressed in terms of its initial state $\rho(0)$ as: $$\rho(t) = \hat{U}(t) \rho(0) \hat{U}^\dagger(t)$$ where $\rho(0)$ is the density operator at time $t=0$. For a pure state initially described by $\rho(0) = \ensuremath{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|}$, its evolution under $\hat{U}(t)$ yields $\rho(t) = \hat{U}(t) \ensuremath{\left|{\psi_0}\middle\rangle\middle\langle{\psi_0}\right|} \hat{U}^\dagger(t) = \ensuremath{\left|{\hat{U}(t)\ensuremath{\left|{\psi_0}\right\rangle}}\middle\rangle\middle\langle{\ensuremath{\left\langle{\psi_0}\right|}\hat{U}^\dagger(t)}\right|} = \ensuremath{\left|{\psi(t)}\middle\rangle\middle\langle{\psi(t)}\right|}$, where $\ensuremath{\left|{\psi(t)}\right\rangle} = \hat{U}(t) \ensuremath{\left|{\psi_0}\right\rangle}$. This shows that the Schrödinger equation for pure states is recovered as a special case within the density operator formalism. The Liouville-von Neumann equation and the density operator formalism are essential for describing the quantum dynamics of both pure and mixed states, particularly in scenarios involving statistical ensembles or open quantum systems. # Conclusion This lecture has provided a comprehensive overview of key concepts in quantum mechanics, starting with the unique nature of quantum measurement and its various interpretations. We explored the probabilistic nature of measurement, the phenomenon of wave function collapse, and interpretations such as the Many-Worlds Interpretation. The discussion extended to different types of measurements, including projective measurements and POVMs, and their role in distinguishing quantum states. We also examined the statistical properties of measurement outcomes, focusing on averagevalues and dispersion. Further, we elucidated the concept of phase in quantum mechanics, distinguishing between the physical irrelevance of global phase and the significance of relative phases. A substantial portion of the lecture was dedicated to quantum entanglement, defining it as a uniquely quantum correlation with implications for non-locality, as evidenced by Bell's Theorem. Superdense coding was presented as a practical application showcasing the power of entanglement in quantum communication. Finally, we introduced the density operator as a powerful tool for describing mixed quantum states and statistical ensembles, and discussed its time evolution under the Liouville-von Neumann equation. This collective understanding forms a robust foundation for further exploration into advanced quantum topics and technologies. **Key Takeaways:** - Quantum measurement is probabilistic and fundamentally alters the state of the system, leading to wave function collapse. - Entanglement represents non-classical correlations that defy classical explanations and underpin many quantum technologies. - Relative phase is a physically significant quantity in quantum mechanics, crucial for interference and quantum computation. - The density operator provides a general framework for describing both pure and mixed quantum states and their time evolution. **Further Topics:** Building upon the concepts introduced in this lecture, future discussions will delve into: - A more in-depth analysis of various interpretations of quantum measurement and their implications. - Detailed exploration of Bell's Theorem, Bell inequalities, and experimental validations of quantum non-locality. - Applications of entanglement in advanced quantum computing algorithms and quantum cryptography protocols. - Quantum operations and quantum channels within the density operator formalism, essential for understanding quantum information processing. - Open quantum systems and decoherence, utilizing the density operator framework to describe realistic quantum systems interacting with their environment. These forthcoming topics will expand on the foundational knowledge established here, further illuminating the profound and transformative nature of quantum mechanics and its applications in quantum technologies.